It is not always necessary to compute derivatives directly from the definition. Instead, several rules have been developed for finding derivatives without having to use the definition directly. These formulas greatly simplify the task of differentiation.

Definition of the Power Rule

The Power Rule of Derivatives gives the following:

For any real number n,

the derivative of f(x) = xn is f ’(x) = nxn-1

which can also be written as

Example:

Differentiate the following:
a) f(x) = x5b) y = x100c) y = t6

Solution:

a) f’’(x) = 5x4

b) y’ = 100x99

c) y’ = 6t5

Derivative of the function f(x) = x

Using the power rule formula, we find that the derivative of the function f(x) = x would be one.

The derivative of f(x) = x is f ’(x) = 1

which can also be written as

Example:

Differentiate f(x) = x

Solution:

f ’(x) = f ’(x1) = 1x0 = 1

Derivative of a Constant Function

Using the power rule formula, we find that the derivative of a function that is a constant would be zero.

For any constant c,

The derivative of f(x) = c is f ’(x) = 0

which can also be written as

Example:

Differentiate the following:
a) f(3)
b) f(157)

Solution:

a) f ‘(3) = f ‘(3x0) = 0(3x-1) = 0
b) f ‘(157) = 0

The Constant Multiple Rule

The constant multiple rule says that the derivative of a constant value times a function is the constant times the derivative of the function.

If c is a constant and f is a differentiable function, then

Example:

Differentiate the following:
a) y = 2x4
b) y = –x

Solution:

The Sum Rule

The Sum Rule tells us that the derivative of a sum of functions is the sum of the derivatives.

If f and g are both differentiable, then

The Sum Rule can be extended to the sum of any number of functions.
For example (f + g + h)’ = f’ + g’ + h’

Example:

Differentiate 5x2 + 4x + 7

Solution:

The Difference Rule

The Difference Rule tells us that the derivative of a difference of functions is the difference of the derivatives.

If f and g are both differentiable, then

Example:

Differentiate x8 – 5x2 + 6x

Solution:

Proof of the Power Rule for Derivatives
An explanation and some examples.

Derivative Calculator

The following derivative calculator can show you the steps and rules used to get the derivative of the given function. Use it to check your answers.

Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.